Problem: Kana is practicing her golf swing at the driving range. Her first hit can be described by the displacement (distance and direction) vector ${\vec{d_1}} = (55, 5)$. Her second hit can be described by the displacement vector ${\vec{d_2}} = (75, 10)$. (Values above are given in meters.) How much farther did Kana's second hit go than her first hit?
Solution: The distance each golf ball went is the magnitude of its displacement vector. Therefore, the expression $\| {\vec{d_2}} \| - \| {\vec{d_1}} \|$ tells us how much farther the second ball went than the first. $\| {\vec{d_2}} \| - \| {\vec{d_1}} \| \approx 20.4$ meters Note: We can find the magnitude of any vector $\vec v$ using the Pythagorean theorem $\| \vec v \|^2 = x^2 + y^2$, where $x$ and $y$ are the components of $\vec v$. For the second question, let's find how far the first golf ball is from the second. To do that, let's think about the horizontal and vertical components of each vector. Horizontally, both balls went to the right, but the second ball went $20$ meters farther. Therefore, the first ball is $20$ meters to the left of the second ball. Vertically, both balls went in the upwards direction, but the second ball went $5$ meters farther. Therefore, the first ball is $5$ meters downwards from the second ball. Note that what we've just done was subtract the vectors component-wise: ${\vec{d_1}} - {\vec{d_2}} = {(55,5)} - {(75,10)} = (-20, -5)$ Performing the subtraction the other way would have resulted in $(20,5)$. This is the distance the second ball is from the first. This means that to model the displacement between the golf balls, we need to subtract one of the vectors from the other. Applying the Pythagorean theorem, we can find the magnitudes of $(-20,-5)$ or $(20,5)$, which are the same. That is, we can find how far the golf balls are apart. $\| {\vec{d_1}} - {\vec{d_2}} \| = \| {\vec{d_2}} - {\vec{d_1}} \| \approx 20.6 \,\text{m}$. Kana's second hit went $20.4$ meters farther than her first hit. The golf balls are $20.6$ meters apart.